The alternative characterizations of inner capacities and inner capacitary measures thereby obtained are given in particular by Theorems REF , REF , REF , REF , REF , REF  and Corollaries REF , REF .
The analysis performed is substantially based on the close interaction between the theory of capacities and that of balayage.
(Concerning the theory of balayage on a locally compact space \(X\)  with respect to a suitable kernel \(\kappa \) , see the author's recent paper [1]}; see also Sects. ,  below for further relevant results, in particular Theorems REF , REF  for a number of alternative characterizations of inner and outer swept measures.)
